A family of non-periodic tilings of the plane by right golden triangles

TL;DR

This paper explores a family of non-periodic tilings of the plane using right golden triangles, providing an alternative, more intuitive proof of their sofic nature through decorated tiles and simpler local rules.

Contribution

It offers a new, more intuitive proof that these tilings are sofic, using more decorated tiles and simpler local rules than previous work.

Findings

The tilings are sofic, derived from local rules.

An alternative proof simplifies understanding of the tilings.

Decorations used in the proof are more intuitive.

Abstract

We study a family of substitution tilings with similar right triangles of two sizes which is obtained using the substitution rule introduced in [Danzer, L. and van Ophuysen, G. A species of planar triangular tilings with inflation factor $\sqrt{-\tau}$. Res. Bull. Panjab Univ. Sci. 2000, 50, 1-4, pp. 137--175 (2001)]. In that paper, it is proved this family of tilings can be obtained from a local rule using decorated tiles. That is, that this family is \emph{sofic}. In the present paper, we provide an alternative proof of this fact. We use more decorated tiles than Danzer and van Ophuysen (22 in place of 10). However, our decoration of supertiles is more intuitive and our local rule is simpler.